Stability of approximate group actions: uniform and probabilistic
Oren Becker
University of Cambridge, UKMichael Chapman
The Hebrew University, Jerusalem, Israel
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Abstract
We prove that every uniform approximate homomorphism from a discrete amenable group into a symmetric group is uniformly close to a homomorphism into a slightly larger symmetric group. That is, amenable groups are uniformly flexibly stable in permutations. This answers affirmatively a question of Kun and Thom and a slight variation of a question of Lubotzky. We also give a negative answer to Lubotzky’s original question by showing that the group is not uniformly strictly stable. Furthermore, we show that , , is uniformly flexibly stable, but the free group , , is not. We define and investigate a probabilistic variant of uniform stability that has an application to property testing.
Cite this article
Oren Becker, Michael Chapman, Stability of approximate group actions: uniform and probabilistic. J. Eur. Math. Soc. 25 (2023), no. 9, pp. 3599–3632
DOI 10.4171/JEMS/1267